metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4⋊4D18, Q8⋊5D18, C36.51D4, C36.18C23, D36.11C22, D4⋊D9⋊6C2, C4○D4⋊3D9, C9⋊C8⋊4C22, C9⋊5(C8⋊C22), (C2×C18).9D4, (C2×D36)⋊10C2, C3.(D4⋊D6), Q8⋊2D9⋊6C2, (C3×D4).33D6, C18.60(C2×D4), (C2×C12).69D6, (C2×C4).21D18, (D4×C9)⋊4C22, (C3×Q8).57D6, (Q8×C9)⋊4C22, C4.25(C9⋊D4), C4.Dic9⋊10C2, C4.18(C22×D9), (C2×C36).46C22, C12.57(C22×S3), C22.6(C9⋊D4), C12.113(C3⋊D4), (C9×C4○D4)⋊1C2, C2.24(C2×C9⋊D4), (C3×C4○D4).12S3, (C2×C6).8(C3⋊D4), C6.108(C2×C3⋊D4), SmallGroup(288,160)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4⋊D18
G = < a,b,c,d | a4=b2=c18=d2=1, bab=dad=a-1, ac=ca, cbc-1=a2b, dbd=a-1b, dcd=c-1 >
Subgroups: 524 in 102 conjugacy classes, 38 normal (30 characteristic)
C1, C2, C2 [×4], C3, C4 [×2], C4, C22, C22 [×5], S3 [×2], C6, C6 [×2], C8 [×2], C2×C4, C2×C4, D4, D4 [×4], Q8, C23, C9, C12 [×2], C12, D6 [×4], C2×C6, C2×C6, M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, D9 [×2], C18, C18 [×2], C3⋊C8 [×2], D12 [×3], C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C22×S3, C8⋊C22, C36 [×2], C36, D18 [×4], C2×C18, C2×C18, C4.Dic3, D4⋊S3 [×2], Q8⋊2S3 [×2], C2×D12, C3×C4○D4, C9⋊C8 [×2], D36 [×2], D36, C2×C36, C2×C36, D4×C9, D4×C9, Q8×C9, C22×D9, D4⋊D6, C4.Dic9, D4⋊D9 [×2], Q8⋊2D9 [×2], C2×D36, C9×C4○D4, D4⋊D18
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], C2×D4, D9, C3⋊D4 [×2], C22×S3, C8⋊C22, D18 [×3], C2×C3⋊D4, C9⋊D4 [×2], C22×D9, D4⋊D6, C2×C9⋊D4, D4⋊D18
►On 72 points
(1 33 15 24)(2 34 16 25)(3 35 17 26)(4 36 18 27)(5 28 10 19)(6 29 11 20)(7 30 12 21)(8 31 13 22)(9 32 14 23)(37 66 46 57)(38 67 47 58)(39 68 48 59)(40 69 49 60)(41 70 50 61)(42 71 51 62)(43 72 52 63)(44 55 53 64)(45 56 54 65)
(1 63)(2 55)(3 65)(4 57)(5 67)(6 59)(7 69)(8 61)(9 71)(10 58)(11 68)(12 60)(13 70)(14 62)(15 72)(16 64)(17 56)(18 66)(19 47)(20 39)(21 49)(22 41)(23 51)(24 43)(25 53)(26 45)(27 37)(28 38)(29 48)(30 40)(31 50)(32 42)(33 52)(34 44)(35 54)(36 46)
(1 2 3 4 5 6 7 8 9)(10 11 12 13 14 15 16 17 18)(19 20 21 22 23 24 25 26 27)(28 29 30 31 32 33 34 35 36)(37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54)(55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72)
(1 4)(2 3)(5 9)(6 8)(10 14)(11 13)(15 18)(16 17)(19 32)(20 31)(21 30)(22 29)(23 28)(24 36)(25 35)(26 34)(27 33)(37 63)(38 62)(39 61)(40 60)(41 59)(42 58)(43 57)(44 56)(45 55)(46 72)(47 71)(48 70)(49 69)(50 68)(51 67)(52 66)(53 65)(54 64)
G:=sub<Sym(72)| (1,33,15,24)(2,34,16,25)(3,35,17,26)(4,36,18,27)(5,28,10,19)(6,29,11,20)(7,30,12,21)(8,31,13,22)(9,32,14,23)(37,66,46,57)(38,67,47,58)(39,68,48,59)(40,69,49,60)(41,70,50,61)(42,71,51,62)(43,72,52,63)(44,55,53,64)(45,56,54,65), (1,63)(2,55)(3,65)(4,57)(5,67)(6,59)(7,69)(8,61)(9,71)(10,58)(11,68)(12,60)(13,70)(14,62)(15,72)(16,64)(17,56)(18,66)(19,47)(20,39)(21,49)(22,41)(23,51)(24,43)(25,53)(26,45)(27,37)(28,38)(29,48)(30,40)(31,50)(32,42)(33,52)(34,44)(35,54)(36,46), (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18)(19,20,21,22,23,24,25,26,27)(28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54)(55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72), (1,4)(2,3)(5,9)(6,8)(10,14)(11,13)(15,18)(16,17)(19,32)(20,31)(21,30)(22,29)(23,28)(24,36)(25,35)(26,34)(27,33)(37,63)(38,62)(39,61)(40,60)(41,59)(42,58)(43,57)(44,56)(45,55)(46,72)(47,71)(48,70)(49,69)(50,68)(51,67)(52,66)(53,65)(54,64)>;
G:=Group( (1,33,15,24)(2,34,16,25)(3,35,17,26)(4,36,18,27)(5,28,10,19)(6,29,11,20)(7,30,12,21)(8,31,13,22)(9,32,14,23)(37,66,46,57)(38,67,47,58)(39,68,48,59)(40,69,49,60)(41,70,50,61)(42,71,51,62)(43,72,52,63)(44,55,53,64)(45,56,54,65), (1,63)(2,55)(3,65)(4,57)(5,67)(6,59)(7,69)(8,61)(9,71)(10,58)(11,68)(12,60)(13,70)(14,62)(15,72)(16,64)(17,56)(18,66)(19,47)(20,39)(21,49)(22,41)(23,51)(24,43)(25,53)(26,45)(27,37)(28,38)(29,48)(30,40)(31,50)(32,42)(33,52)(34,44)(35,54)(36,46), (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18)(19,20,21,22,23,24,25,26,27)(28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54)(55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72), (1,4)(2,3)(5,9)(6,8)(10,14)(11,13)(15,18)(16,17)(19,32)(20,31)(21,30)(22,29)(23,28)(24,36)(25,35)(26,34)(27,33)(37,63)(38,62)(39,61)(40,60)(41,59)(42,58)(43,57)(44,56)(45,55)(46,72)(47,71)(48,70)(49,69)(50,68)(51,67)(52,66)(53,65)(54,64) );
G=PermutationGroup([(1,33,15,24),(2,34,16,25),(3,35,17,26),(4,36,18,27),(5,28,10,19),(6,29,11,20),(7,30,12,21),(8,31,13,22),(9,32,14,23),(37,66,46,57),(38,67,47,58),(39,68,48,59),(40,69,49,60),(41,70,50,61),(42,71,51,62),(43,72,52,63),(44,55,53,64),(45,56,54,65)], [(1,63),(2,55),(3,65),(4,57),(5,67),(6,59),(7,69),(8,61),(9,71),(10,58),(11,68),(12,60),(13,70),(14,62),(15,72),(16,64),(17,56),(18,66),(19,47),(20,39),(21,49),(22,41),(23,51),(24,43),(25,53),(26,45),(27,37),(28,38),(29,48),(30,40),(31,50),(32,42),(33,52),(34,44),(35,54),(36,46)], [(1,2,3,4,5,6,7,8,9),(10,11,12,13,14,15,16,17,18),(19,20,21,22,23,24,25,26,27),(28,29,30,31,32,33,34,35,36),(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54),(55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72)], [(1,4),(2,3),(5,9),(6,8),(10,14),(11,13),(15,18),(16,17),(19,32),(20,31),(21,30),(22,29),(23,28),(24,36),(25,35),(26,34),(27,33),(37,63),(38,62),(39,61),(40,60),(41,59),(42,58),(43,57),(44,56),(45,55),(46,72),(47,71),(48,70),(49,69),(50,68),(51,67),(52,66),(53,65),(54,64)])
51 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 8A | 8B | 9A | 9B | 9C | 12A | 12B | 12C | 12D | 12E | 18A | 18B | 18C | 18D | ··· | 18L | 36A | ··· | 36F | 36G | ··· | 36O |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 8 | 8 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 12 | 18 | 18 | 18 | 18 | ··· | 18 | 36 | ··· | 36 | 36 | ··· | 36 |
| size | 1 | 1 | 2 | 4 | 36 | 36 | 2 | 2 | 2 | 4 | 2 | 4 | 4 | 4 | 36 | 36 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
51 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D6 | D9 | C3⋊D4 | C3⋊D4 | D18 | D18 | D18 | C9⋊D4 | C9⋊D4 | C8⋊C22 | D4⋊D6 | D4⋊D18 |
| kernel | D4⋊D18 | C4.Dic9 | D4⋊D9 | Q8⋊2D9 | C2×D36 | C9×C4○D4 | C3×C4○D4 | C36 | C2×C18 | C2×C12 | C3×D4 | C3×Q8 | C4○D4 | C12 | C2×C6 | C2×C4 | D4 | Q8 | C4 | C22 | C9 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 2 | 2 | 3 | 3 | 3 | 6 | 6 | 1 | 2 | 6 |
Matrix representation of D4⋊D18 ►in GL4(𝔽73) generated by
| 66 | 59 | 0 | 0 |
| 14 | 7 | 0 | 0 |
| 48 | 20 | 7 | 14 |
| 53 | 28 | 59 | 66 |
| 31 | 36 | 4 | 31 |
| 37 | 68 | 42 | 46 |
| 50 | 38 | 42 | 37 |
| 35 | 12 | 36 | 5 |
| 28 | 70 | 0 | 0 |
| 3 | 31 | 0 | 0 |
| 25 | 45 | 45 | 3 |
| 28 | 53 | 70 | 42 |
| 45 | 42 | 0 | 0 |
| 70 | 28 | 0 | 0 |
| 0 | 0 | 66 | 59 |
| 0 | 0 | 66 | 7 |
G:=sub<GL(4,GF(73))| [66,14,48,53,59,7,20,28,0,0,7,59,0,0,14,66],[31,37,50,35,36,68,38,12,4,42,42,36,31,46,37,5],[28,3,25,28,70,31,45,53,0,0,45,70,0,0,3,42],[45,70,0,0,42,28,0,0,0,0,66,66,0,0,59,7] >;
D4⋊D18 in GAP, Magma, Sage, TeX
D_4\rtimes D_{18} % in TeX
G:=Group("D4:D18"); // GroupNames label
G:=SmallGroup(288,160);
// by ID
G=gap.SmallGroup(288,160);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,254,219,675,185,80,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^18=d^2=1,b*a*b=d*a*d=a^-1,a*c=c*a,c*b*c^-1=a^2*b,d*b*d=a^-1*b,d*c*d=c^-1>;
// generators/relations